During the application of the conventional textbook SP approach to
certain many-particle quantum systems the convergence of variational
methods may become prohibitively slow. In these cases the
Hamiltonian is, typically, a complicated partial differential
operator \(\mathfrak {h}\)  acting in a complicated many-body Hilbert
space \({\cal L}=L^2(\mathbb {R}^d)\)  with a large
dimension-representing exponent \(d\) . The convergence becomes
particularly slow when the particles are fermions. In this case,
indeed, the system has to obey the Pauli exclusion principle
[1]} so that “it has become customary to map Hermitian
fermion operators onto non-Hermitian boson operators” [2]}.
Purely empirically it has been revealed, originally by Dyson
[3]}, that the calculations (typically, of the low-lying
spectra) may perceivably be accelerated via a judicious a
priori  simulation of the correlations. In the language of
mathematics this means that it makes sense to replace the initial
“fermionic” Hilbert space \({\cal L}\)  by an auxiliary (and,
formally, non-equivalent) “bosonic” Hilbert space, say, \({\cal H}_{math}\) . In the latter (and, by assumption,
user-friendlier) space the system becomes described by a new
Hamiltonian (say, \(H\) ). Naturally, a successful (often, just
intuition-supported) estimate and simulation of the structure of the
correlations of fermions has often been found to imply a decisive
acceleration of the convergence of the variational treatment of the
bosonic operator \(H\) . In the language of numerical mathematics one
may speak about an isospectral preconditioning (REF )
of the Hamiltonian.
